Closure to “Effect of Seepage-Induced Nonhydrostatic Pressure Distribution on Bed-Load Transport and Bed Morphodynamics” by Simona Francalanci, Gary Parker, and Luca Solari

DISCUSSIONS AND CLOSURES

Discussion of “Transition between Two

Bed-Load Transport Regimes: Saltation

and Sheet-Flow” by P. Gao

March 2008, Vol. 134, No. 3, pp. 340–349.

DOI: 10.1061/共ASCE兲0733-9429共2008兲134:3共340兲

Benoît Camenen1

1_{Cemagref UR HHLY, 3 bis quai Chauveau, CP 220, 69336 LYON cedex}

09, France. E-mail: benoit.camenen@cemagref.fr

The author presented an interesting theoretical model for predict-ing the onset of the sheet-flow regime. It introduces the parameter

P_b, which characterizes the proportion of grains moved from the surface layer. The proposed relationship for Pb is however de-rived from the bed-load formula by Abrahams and Gao 共2006兲, which yields a nearly constant Shields parameter␪t⬇0.5, a curi-ous result that merits discussion.

Model for Bed-Load Transport

From the hypothesis proposed by the author, Pbis directly related to the volumetric bed-load transport rate qband the mean grain velocity Ub关Eq. 共3兲 in the paper兴. As all further developments to estimate qband Ubare based on the work by Abrahams and Gao 共2006兲, the proposed equation for Pbappears to be a different way to write the bed-load formula originally proposed by Abrahams and Gao. Using the dimensionless bed load transport ⌽ and the definition of the Shields parameter ␪=RhS/关共s−1兲D兴=u_*2/关共s − 1兲gD兴 共see Gao’s paper for the notations; S is introduced here for the energy slope, Rh is the hydraulic radius, u_* is the shear velocity, and s =␳s/␳ is the relative density of the sediment兲, Eqs. 共10兲 and 共12兲 in the paper yield

⌽b= q_b

冑

共s − 1兲gD3= C₀P_bU_b

冑

共s − 1兲gD= ␪u

冑

共s − 1兲gD

冉

1 − ␪c ␪

冊

3.4 =␪3/2u u * G3.4 _共1兲

which is exactly the equation proposed by Abrahams and Gao 关2006, Eq. 共25兲兴.

Abrahams and Gao 共2006兲 argued that the excess bed shear stress is balanced by intergranular collisions force and fluid drag/ lift force, which may be written in a dimensionless form: ␪−␪c =␪tg+␪tf. Bagnold共1966兲 introduced a bed-load transport model where most of the energy is used to support the intergranular collisions force共␪cand␪tfassumed to be negligible兲

⌽b= e_b tan␣␪ 3/2u u * 共2兲 where eb= bed-load efficiency. The angle of repose tan␣ comes from the relationship ␪tg= tan␣W⬘/关共␳s−␳兲gD兴 where W⬘ is the immersed weight of the load. It explains the coefficient 0.6 in Gao’s paper共the angle of repose for sands is usually taken as ␣

⬇30°兲. Thus, Gao’s developments are summarized as

P_b= ␪tg

C₀tan␣ while Ub=

q_btan␣ ␪tgD

共3兲 For very low rates,␪cis not negligible and␪tfⰇ␪tg共Bridge and Bennett 1992兲. When ␪ is close to ␪c,␪tgⰆ␪ and Pb→0; when ␪Ⰷ␪c,␪tg⬇␪ and Pb= 1共inception of the sheet flow layer兲 yields thus to␪tg⬇C0tan␣=0.4. A link with Gao’s results may be ob-tained using the empirical relationships suggested by Abrahams and Gao 共2006兲: ␪tg/共␪−␪c兲=G2 and Eq. 共12兲 for Ub. In that sense, Pbmainly expresses the effects of the critical Shields pa-rameter on bed-load transport but not the transition between sal-tation and sheet flow regimes.

Effect of the Critical Shields Parameter on Bed-Load Transport

Estimating bed-load transport is challenging when␪cis not neg-ligible and␪−␪c⬇␪tf. Following Meyer-Peter and Müller共1948兲, most authors add a factor that varies with the ratio rt=␪cr/␪ and assume qs= 0 when␪⬍␪cr. Camenen and Larson共2005兲 argued that the estimation of␪citself is subject to uncertainties, and then suggested an exponential function of␪/␪cr共see Table 1兲.

For the experimental study used by Gao, side wall effects should have a significant influence on the estimation of the total shear stress applied to the bed as the width of the channel is narrow共b=0.1 m兲 and velocities are large 共0.8⬍u⬍2.4 m/s兲. As Gao seems to have used Rh= h for his computations, the same assumption has been used below, but these results may differ significantly if a side wall correction is taken into account 共cf. method of Vanoni and Brooks 1957兲.

In Table 1, the studied bed-load formulas are tested against Gao’s data, and for each of these formulas, the ratio ⌽/␪3/2 is plotted versus the ratio␪/␪cr= 1/rtin Fig. 1. The performance of the Abrahams and Gao共2006兲 formula is weaker than those pro-vided by the Meyer-Peter and Müller 共1948兲 or Camenen and Larson共2005兲 formulas. The Abrahams and Gao formula tends to overestimate bed load rates for the data with D = 1.16 mm and underestimate bed load rates for the data with D = 7 mm. As the equation for Pbis closely related to the Abrahams and Gao for-mula, its prediction capability may also be uncertain. The effect of the critical Shields parameter is also sharper using the Abra-hams and Gao formula. rtvarying from 3 to 1.5 yields a decrease of bed load transport by a factor 10.6. For the Meyer-Peter and Müller and Camenen and Larson formulas, it yield a decrease by a factor 2.8 and 4.5, respectively.

Onset of the Sheet Flow Regime

For a fixed bed or very low bed-load rates, the flow resistance is generally related to the size of the bed material such as the rough-ness height ks⬇2D 共see Camenen et al. 2006兲. For the sheet flow regime, Wilson共1966兲 observed a linear relationship between ks and ␪. Sumer et al. 共1996兲 observed however a more complex relationship where ksis function of both D and␪. Camenen et al.

共2006兲 and Recking et al. 共2008兲 showed that ksmay be a func-tion of␪ or of qbnot only in the sheet- flow regime but also for relatively intense bed load rates. Camenen et al.共2006兲 suggested a critical Shields parameter ␪cr,ur, up to which the ratio ks/D increases with␪ 共see Fig. 2兲. ␪cr,urwas found to be a function of the Froude number and a dimensionless settling velocity. In Fig. 2, ks, calculated based on Nikuradse’s resistance relationship共cf. Camenen et al. 2006兲, is plotted against ␪ using Gao’s experi-ments. For D = 7 mm, there is an increase of ks/D with ␪, which is

roughly predicted by the Camenen et al. model. On the contrary, for D = 1.16 mm, ks/D is a decreasing function of ␪. How could it be explained? For this material, Gao observed a value for ␪t ⬇0.5. And for ␪⬎␪t, he observed an increasing thickness␦sof the sheet flow layer with␪. As ks⬇␦s共Wilson 1966兲, these results seem surprising.

Although the theoretical aspects of the model suggested by Gao are of much interest, the final relationship presents some weaknesses. The model yields a nearly constant value for the onset of the sheet flow regime共␪t⬇0.5兲, whereas values found in the literature are often larger共0.8⬍␪t⬍1.0 for sheet flow regime with fine sediments兲. It has not been validated with other data sets with different grain sizes, and looking at his own data with D = 7 mm, one may wonder if the model is valid, even if no sheet flow was apparently observed. The suggested formula for Pb sig-nificantly depends on the bed load model by Abrahams and Gao, which yields relatively poor results for the prediction of the bed load rates. Lastly, this model hypothesizes a direct transition be-tween saltation and sheet-flow regimes without bed forms occur-ring, which seems unrealistic共at least for fine sands兲. Thus, there are several reasons to believe this model may be of limited appli-cability.

References

Abrahams, A., and Gao, P.共2006兲. “A bed-load formula transport model for rough turbulent open-channel flows on plane beds.” Earth Surf.

Processes Landforms, 31共7兲, 910–928.

Bagnold, R. 共1966兲. “An approach of sediment transport model from general physics.” Technical Report 422-1, U.S. Geol. Survey Prof. Paper, Reston, Va.

Bridge, J., and Bennett, S.共1992兲. “A model for the entrainment and transport of sediment grains of mixed sizes, shapes, and densities.”

Water Resour. Res., 28共2兲, 337–363.

Camenen, B., Bayram, A., and Larson, M.共2006兲. “Equivalent roughness height for plane bed under steady flow.” J. Hydraul. Eng., 132共11兲, 1146–1158.

Camenen, B., and Larson, M. 共2005兲. “A bedload sediment transport formula for the nearshore.” Estuarine Coastal Shelf Sci., 63共1–2兲, 249–260.

Meyer-Peter, E., and Müller, R.共1948兲. “Formulas for bed-load trans-port.” Proc., 2nd Meeting of the International Association for

Hydrau-lic Structures Research, pp. 39–64, IAHR, Delft, The Netherlands.

Recking, A., Frey, P., Paquier, A., Belleudy, P., and Champagne, J. 共2008兲. “Bed-load transport flume experiments on steep slopes.” J.

Hydraul. Eng., 134共9兲, 1302–1310.

Sumer, B., Kozakievicz, A., Fredsœ, J., and Deigaard, R.共1996兲. “Veloc-ity and concentration profiles in the sheet-flow layer of movable bed.”

J. Hydraul. Eng., 122共10兲, 549–558.

Vanoni, V., and Brooks, N.共1957兲. “Laboratory studies of the roughness and suspended load of alluvial streams.” Technical Rep., Sedimenta-tion Lab., California Institute of Technology, Pasadena, Calif. Wilson, K. 共1966兲. “Bed-load transport at high shear stress.” J. Hydr.

Div., 92共11兲, 49–59.

Table 1. Relationships for the Effect of the Critical Shields Parameter on the Bed-load Transport Rate and Statistical Results Compared to the

Experimental Data by Guo

Author共s兲 Meyer-Peter & Müller Abrahams & Gao Camenen & Larson

⌽/␪3/2 _8共1−r

t兲3/2 u/u_*共1−rt兲3.4 12 exp共−4.5rt兲

p_⫾1.5共%兲 79 38 79

mean关log共⌽pred./⌽meas.兲兴 −0.092 +0.100 +0.007

std关log共⌽pred./⌽meas.兲兴 0.37 0.72 0.31

Note: p_⫾1.5corresponds to the percentage of data correctly predicted within a factor 1.5 allowed.

Fig. 1. Effect of the critical Shields parameter on the bed load

trans-port rate using the Meyer-Peter and Müller, Abrahams and Gao, and Camenen and Larson formulas

10−1 100 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 θ k s /D Gao data, D=7mm Gao data, D=1.16mm Eq. Camenen et al, D=7mm Eq. Camenen et al, D=1.16mm

Fig. 2. Estimation of the roughness ratio k_s/D as a function of the

Closure to “Transition between Two

Bed-Load Transport Regimes: Saltation

and Sheet Flow” by Peng Gao

March 2008, Vol. 134, No. 3, pp. 340–349.

DOI: 10.1061/共ASCE兲0733-9429共2008兲134:3共340兲

Peng Gao1

1_{Dept. of Geography, Syracuse Univ., Syracuse, NY 13244.}

The writer thanks the discusser for raising several issues regard-ing the proposed model describregard-ing the transition between two bed-load transport regimes. It appears that further clarification on the results of this model is needed.

It is agreed that the theoretical expression for Pb关i.e., Eq. 共7兲 in the original paper兴 was partly derived from the bed-load trans-port equation developed by Abrahams and Gao共2006兲, hereafter referred to as the AG equation. However, the original Eq. 共7兲 is far more than a different version of the AG equation. First, Pbhas a clear sedimentological meaning. It represents the proportion of grains in the top layer of the original mobile bed that is trans-ported as bed load. Second, the original Eq.共7兲 is only valid for a limited range of values of␪ because Pbby definition cannot be greater than 1. In contrast, the AG equation applies to flows with any possible value of␪. Once Pb= 1, the corresponding␪ value marks the hydraulic transition from the saltation to sheet-flow regime. This is the key theoretical idea demonstrated in the origi-nal paper and cannot be derived from the AG equation.

The discusser’s Eq. 共3兲 may be alternately derived in two steps. Combining the original Eq.共7兲 with eb= Ub/uG and G=1 −␪c/␪ 共Abrahams and Gao 2006兲, and replacing qb by the sub-merged bed-load transport rate ib= qb共␳s−␳兲g lead to

Pb=

i_b

冉

1 −␪c ␪

冊

ebC0D共␳s−␳兲gu

共4兲 Further combination of Eq. 共4兲 with both Bagnold’s energy equation ibtan␣=eb␻ 共Bagnold 1966; 1973兲 and the definition of stream power␻=␶u gives rise to

P_b= ␪ − ␪c

C₀tan␣ 共5兲

Comparison of Eq.共5兲 with the discusser’s Eq. 共3兲 indicates that the latter is a simplified form with an implied assumption that ␪tgⰇ␪tf. However, this assumption is only valid for high flow rates共i.e., ␪Ⰷ␪c兲 under which bed-load grains are primarily sup-ported by grain-to-grain collisions rather than by turbulent lift. Therefore, the discusser’s concern is not justified for low flow rates. Additionally, when ␪Ⰷ␪c, the discusser’s conclusions that ␪tg⬇␪ and Pb= 1 are erroneous, because the condition␪tg⬇␪ is only valid when flows have extremely high ␪ values, which are greater than the threshold value of ␪ for the transition from sal-tation to sheet-flow regimes, ␪t. Thus, neither Eq. 共5兲 nor the discusser’s Eq.共3兲 is valid for flows with high ␪ values.

Moreover, the discusser’s conclusion ␪t= C0tan␣=0.4 as-sumes tan␣=0.6. Abrahams and Gao 共2006兲 showed, however, that this assumption is only valid for ␪Ⰷ␪t. Therefore, use of tan␣=0.6 to calculate ␪tis inappropriate.

The discusser further questions Pbby stating that it only re-flects the effect of␪con bed-load transport. The proposed model for the transitional condition关i.e., Eq. 共11兲 in the original paper兴 should be correctly used by setting Pb= 1 and then calculating␪.

Table 1. Values of␪tfor Different␪c

␪c 0.01 0.02 0.03 0.04 0.05 0.06

␪t 0.42 0.45 0.48 0.5 0.53 0.55

Because the original Eq. 共11兲 contains ␪c, the calculated ␪t is affected by the value of ␪c, which is demonstrated in Table 1. When ␪c increases from 0.03 to 0.06 共i.e., 100%兲, ␪t merely changes from 0.48 to 0.55 共i.e., 17%兲. Therefore, any errors in determining ␪c 共Buffington and Montgomery 1997兲 will have a limited effect on␪t.

The discusser attributed the inaccuracy of the AG equation to the effect of␪c on bed-load transport by suggesting in Fig. 1 in the discussion that this equation has the worst performance com-pared to the other two equations hereafter denoted as MPM 共1948兲 and CL 共2006兲. This comparison was based, however, on only a single data set, while the AG equation was developed using a group of data sets representing a wide range of hydraulic and sedimentological conditions关the original Eq. 共11兲 was also devel-oped from multiple data sets covering a broad range of hydraulic and sedimentological conditions兴. A more broad-based compari-son of predicted␾ with the measured ␾ for the three equations selected by the discusser may be undertaken using the data sets complied in the original paper. Fig. 1 shows these results in which the data labeled Gao 共2003兲 refer to those complied from pub-lished resources described in detail in Gao 共2003兲. Although the CL equation predicts bed load transport rates better than the AG equation for data with grain size of 1.16 mm, the AG equation performs significantly better than either CL or MPM equations for the remainder of the large data set.

Finally, the discusser challenged the original Eq.共11兲 by argu-ing that the dimensionless roughness ks/D should increase with ␪, whereas for data with grain size of 1.16 mm, it decreases with␪ 共Fig. 2 in the discussion兲, which contradicts the empirical obser-vation of bed-load layer thickness reported in the original paper. This conclusion is incorrect. First, the inverse relationship be-tween ks/D and ␪ exists in the sheet-flow regime 共Fig. 2 in the discussion兲 wherein the discusser agrees that ksis approximately equivalent to the bed-load layer thickness ␦s. However, several studies共Cheng 2003; Pugh and Wilson 1999; Sumer et al. 1996; Wilson 1989兲 have shown that ␦s/D is a linear function of ␪, which means␦sand hence ksare independent of D in the sheet-flow regime. In contrast, ksin Fig. 2 in the discussion was calcu-lated using Nikuradse’s resistance equation, which is affected by

D. Therefore, this resistance equation is not valid in the

sheet-flow regime for estimating ks. Second, ksas described in Cam-enen et al. 共2006兲, is influenced by many factors such as grain size and density, setting velocity, and Froude number. Accord-ingly, it should be expected that for a given␪, ksmay have a wide range of numerical values. This is supported by the large scatter of the data in Fig. 2.

References

Bagnold, R. A.,共1973兲. “The nature of saltation and of ‘bed-load’ trans-port in water.” Proc., R. Soc. Lond. A Math. Phys. Sci., 332, 473–504. Buffington, J. M., and Montgomery, D. R.,共1997兲. “A systematic analysis of eight decades of incipient motion studies, with spatial reference to gravel-bedded rivers.” Water Resour. Res., 33, 1993–2029.

Cheng, N.-S., 共2003兲. “A diffusive model for evaluating thickness of bedload layer.” Adv. Water Resour., 26, 875–882.

Discussion of “Effect of Seepage-Induced

Nonhydrostatic Pressure Distribution on

Bed-Load Transport and Bed

Morphodynamics” by Simona Francalanci,

Gary Parker, and Luca Solari

April 2008, Vol. 134, No. 4, pp. 378–389.

DOI: 10.1061/共ASCE兲0733-429共2008兲134:4共378兲

Tom E. Baldock1and Peter Nielsen2

1_{School of Civil Engineering, Univ. of Queensland, Brisbane, QLD 4072,}

Australia.

2_{School of Civil Engineering, Univ. of Queensland, Brisbane, QLD 4072,}

Australia.

The authors present a very interesting and useful paper on the effects of seepage on bed-load sediment transport and bed mor-phodynamics. Importantly, they consider the influence of the seepage on the free stream flow, which has been neglected in many previous studies. However, the discussers would like to note that the topic is not new and that substantial previous experi-mental work has been published in the fluvial and coastal sedi-ment transport literature. The results presented by the authors are both consistent and inconsistent with this previous work. Further-more, their modification to the Shields parameter 关Eq. 共39兲 in their paper兴 appears only partially complete, neglecting the influ-ence of voids on the bulk seepage force that arises during seepage flow through a porous bed. In addition, a similar approach has previously been suggested by Nielsen 共1998兲 and Baldock and Holmes 共1999兲. More important perhaps, the pioneering steady flow work by Martin 共1970兲, which was not considered by the authors, shows that the effects of seepage may have opposite effects depending on the character of the sediment. Nevertheless, the clear morphological differences observed by authors present clear evidence of a seepage effect, but we are not convinced that the full physics of the problem has yet been reconciled with all experimental observations. This paper is an attempt to broaden the discussion of seepage effects on sediment transport within the context of channel flows, uniform oscillatory flows, and nonuni-form wave motion.

Since the work of Martin共1970兲, the other historical develop-ments have been briefly as follows. In steady flows, Watters and Rao 共1971兲 and Willets and Drossos 共1975兲 found that seepage did not effect sediment motion, while Oldenziel and Brink共1974兲 did identify an influence. Löfquist 共1975兲 investigated the pos-sible importance of seepage flow in an oscillatory flow facility but the presence of ripples clouded the conclusions. However, during oscillatory water tunnel flow tests, Kruijt 共1976兲 also found no significant seepage effects on sediment transport. Davis et al. 共1993兲 performed field experiments into the stabilizing effect of lowering the watertable and enhancing downward seepage on beach morphodynamics. The effect was found to be too small to measure in the presence of other morphodynamic processes such as rip cell migration. Oh and Dean共1995兲 investigated the effects of an artificially elevated or lowered watertable behind the beach on beach erosion. The conclusions were unclear or contradictory. Field and laboratory investigations reported by Conley and Inman 共1992, 1994兲 created renewed interest in seepage effects in coastal flows outside the swash zone. Nielsen共1998兲 suggested an adap-tation of the Shields parameter along similar lines to those the authors proposed in their paper, which is discussed further below. Baldock and Holmes 共1999兲 performed laboratory experiments under both steady flows and surface waves with both suction 共seepage into the bed兲 and injection 共seepage out of the bed兲, including fluidized bed conditions. No differences in threshold transport conditions were observed for steady currents, even over fluidized beds. Nielsen et al.共2001兲 performed sediment mobility experiments under waves using a horizontal sand bed, again with both injection and suction. The effects of even very strong seep-age were found to be weak.

The authors note three effects of seepage on sediment trans-port:共1兲 the effect of seepage on bed shear stress; 共2兲 the effect of seepage on immersed particle weight as used in the Shields pa-rameter; and共3兲 the effect of seepage on the threshold of motion, as given by the critical Shields number. The effect of seepage on the shear stress can be further subdivided into two terms, one of which modifies the free stream flow through continuity and one which changes the boundary layer structure. The influence of suc-tion or injecsuc-tion on boundary layer structure has long been known, and applied, in aeronautics共Turcotte 1960兲 and also iden-tified for flows over sediment beds 共Conley and Inman 1992; Chen and Chiew 2004兲. The authors ignore the possible influence of the seepage in directly modifying the bed shear stress in their model. However, very importantly, they include the continuity effect and demonstrate how this leads to strong secondary influ-ences on the sediment transport, in their case through a backwater effect. To the discussers’ knowledge, the continuity effect has not been considered in most of the experimental work cited above, and this neglect may explain some of the conflicting results.

In terms of the effect of seepage on immersed particle weight, this has been quantified by considering the additional nonhydro-static pressure gradient acting within the pore water. However, both Nielsen共1998兲 and the authors neglected the porosity of the sediment within the bed material when determining the seepage force that modifies the effective bulk weight of the sediment. The correct critical hydraulic gradient, i = ic, that induces piping, or fluidization, of a bed of sediment grains with density s and po-rosity n is

i_c= −共s − 1兲共1 − n兲 共1兲

rather than simply ic= −共s−1兲, as is well known in soil mechanics 共Smith 1968兲. The authors use the correct relationship for the critical hydraulic gradient, from Cheng and Chiew共1999兲, in their expression for the critical Shields number 关their Eqs. 共30兲 and

共32兲兴. This is inconsistent with their Eqs. 共16兲 and 共39兲, which do not account for porosity.

Nielsen共1998兲 proposed that the effect of the seepage on the boundary layer and on the immersed sediment weight could be parameterized in the form

␪ = u *o 2

冉

_{1 −␣} v u *0

冊

gd₅₀

冉

s − 1 −␤v K

冊

共2兲 wherev = seepage velocity, positive upward and␤⫽empirical

pa-rameter introduced to account for a possible reduction in the seep-age force on the interfacial particles. Here, the denominator is similar to that proposed by Cheng and Chiew 共1999兲 and the authors, but the potential effect of the seepage on modifying the boundary layer structure is also incorporated in the numerator.

If the porosity is correctly accounted for, then this equation should read ␪ = u *o 2

冉

_{1 −␣} v u *0

冊

gd₅₀共s − 1兲

冉

1 −␤v vf

冊

共3兲

where vf= seepage velocity at incipient fluidization of the sedi-ment bed. An alternative form is

␪ = u *o 2

冉

_{1 −␣} v u *0

冊

gd₅₀共s − 1兲

冉

1 −␤i i_c

冊

共4兲 where the term in the denominator is equivalent to that proposed by Baldock and Holmes共1999兲 for ␤=1. Analogous results apply for the critical Shields parameter. Eqs. 共3兲 and 共4兲 correctly re-duce the effective weight of the grains within the bed to zero for fluidized or piped conditions and reduce to the usual expressions for Shields number in the absence of seepage. Eq.共2兲 above and Eqs.共16兲 and 共39兲 in the original paper do not yield zero effective stress within the bed for i = ic= −v/K⬇−1, which is the usual condition for incipient fluidization of sand beds 共i.e., s=2.65, n = 0.4兲.

Equations of the form of Eqs. 共2兲–共4兲, and equivalent equa-tions in the original paper, which reduce the effective weight of the sediment, become unbounded as the denominator tends to zero, in physical terms as fluidization of the bed occurs共a state of zero effective stress兲. Consequently, using this modification of the Shields parameter, it is expected that for fluidized beds, incipient motion should occur at very low free stream flow rates and very large transport rates would occur under normal free stream flow conditions. However, this is contrary to both direct observation 共Baldock and Holmes 1999兲, and can also be inferred from the data in the authors’ paper. Their experiments show that the scour depths remain finite, even though the critical hydraulic gradient appears to be exceeded during their experiments for Qseepage ⬎140L/s. Baldock and Holmes 共1999兲 performed experiments with two different grain sizes and densities and both strong suc-tion and injecsuc-tion, and observed no significant difference in the steady flow discharge required for incipient motion, consistent with much of previous experimental data cited above. Further-more, incipient motion occurred at the same steady flow condi-tions even if the bed was totally fluidized共i.e., quick conditions兲.

Baldock and Holmes共1999兲 suggested that an explanation for this anomaly could be found by comparing the settling velocity,

w_s, of the sediment particles with the velocity of the flow across the fluid/sediment interface,v. For a sand bed prior to

fluidiza-tion, the maximum possible flow velocity out of the sand bed will be approximately equal to the hydraulic conductivity, K, assum-ing ic⬇−1=−v/K. This flow velocity is generally two orders of magnitude smaller than the settling velocity, ws. However, once a particle lifts out of its bed recess and starts moving, it no longer experiences the seepage force that only acts within the bed matrix. Consequently, the self-weight, as paramterized by the settling ve-locity, dominates over the drag force from the small vertical flow out of the bed. The pressure gradient within the clear fluid is negligible. Essentially, the problem is that the Shields parameter represents a force balance on grains outside of the fluid-sediment matrix, whereas the seepage force acts only within the fluid-sediment matrix.

We therefore propose that the present treatment of this prob-lem by equations of the form of Eq.共39兲 in the authors’ paper or Eqs.共2兲–共4兲 in the present paper is neither particularly convincing with regards the extensive experimental data, nor easily justified on the basis of the fluid physics. Nevertheless, seepage does affect the boundary layer at the very least through the injection or re-moval of fluid mass with different momentum and turbulence characteristics. This may lead to strong indirect effects on sedi-ment transport through changes to suspended sedisedi-ment concentra-tion profiles or changes in bed forms.

The importance of the authors’ paper is that it shows that it is critical to consider secondary effects on the sediment transport rates, and this may explain conflicting data from previous experi-mental studies, and should be carefully considered in future ex-periments. A secondary effect identified by Baldock and Holmes 共1999兲 was that injection promoted ripple formation and changed ripple growth rates on an initially plane bed, while suction re-duced ripple formation and indeed could prevent ripple formation entirely at low flow rates, even though particles continued to move as bed load. Baldock and Holmes 共1999兲 found that changes in ripple regime led to significant changes in roughness and net suspended sediment transport. It would be instructive to know if the authors observed any changes in ripple size or ripple regime over the seepage box in their experiments.

Finally, Baldock and Holmes 共1999兲 suggested that while seepage has little direct influence on saltating bed load particles, it might have a strong influence on sheet flow layers, since the layer may act as a fluid-sediment matrix within which the seepage force can act. Settling velocities are also reduced at high sediment transport concentrations 共Nielsen et al. 2002; Baldock et al. 2004兲, and thus the vertical seepage velocity becomes relatively larger when compared to the particle settling velocity. Such con-ditions occur during the uprush and backwash of wave run-up on beaches, as well as within sheet flows in channel flows.

References

Baldock, T. E., and Holmes, P. 共1999兲. “Seepage effects on sediment transport by waves and currents.” Proc., 26th Int Conf Coastal

Engi-neering, Copenhagen, ASCE, Reston, Va., 3601–3614.

Baldock, T. E., Tomkins, M. R., Nielsen, P., and Hughes, M. G.共2004兲. “Settling velocity of sediments at high concentrations.” Coast. Eng.,

51共1兲, 91–100.

Chen, X., and Chiew, Y. M.共2004兲. “Velocity distribution of turbulent

open-channel flow with bed suction.” J. Hydraul. Eng., 140共92兲, 140– 148.

Cheng, N. S., and Chiew, Y. M.共1999兲. “Incipient motion with seepage.”

J. Hydraul. Res., 37共95兲, 665–681.

Conley, D. C., and Inman, D. L.共1992兲. “Field observations of the fluid-granular boundary layer flow under near breaking waves.” J.

Geo-phys. Res., 97共No C6兲, 9631–9643.

Conley, D. C., and Inman, D. L.共1994兲. “Ventilated oscillatory boundary layers.” J. Fluid Mech., 273, 261–284.

Davis, G. A., Hanslow, D. J., Hibbert, K., and Nielsen P.共1993兲. “Gravity drainage: A new method of beach stabilisation through drainage of the watertable.” Proc., 23rd Int Conf Coastal Eng., ASCE, Reston, Va., 1129–1141.

Francalanci, S., Parker, G., and Solar, L. 共2008兲. “Effect of seepage-induced non-hydrostatic pressure distribution on bed-load transport and bed morphodynamics.” J. Hydraul. Eng., 134共4兲, 378–389. Kruijt, J. A.共1976兲. “On the influence of seepage on incipient motion and

sand transport.” Report No. STF60 AT6046, Tech, Univ., Trondheim, Norway.

Löfquist, K. E. B.共1975兲. “An effect of permeability on sand transport by waves.” Tech Memo 62, U.S. Army Coastal Eng. Res. Center. Martin, C. S.共1970兲. “Effect of a porous sand bed on incipient sediment

motion.” Water Resour. Res., 6共4兲, 1162–1174.

Nielsen, P.共1998兲. “Coastal groundwater dynamics.” Coastal Dynamics

’97, ASCE, Reston, Va., 546–555.

Nielsen, P., Robert, S., Moeller-Christiansen, B., and Oliva, P. 共2001兲. “Infiltration effects on sediment mobility under waves.” Coast. Eng.,

42共2兲, 105–114.

Oh, T.-M., and Dean, R. G.共1995兲. “Effects of controlled water table on beach profile dynamics.” Proc., 24th ICCE, ASCE, 2449–2459. Oldenziel, D. M., and Brink, E. E.共1974兲. J. Hydr. Div., 100共7兲, 935–

949.

Smith, G. N. 共1968兲. Elements of soil mechanics, BSP Professional Books, London.

Turcotte, D. L.共1960兲. “A sub-layer theory for fluid injection into incom-pressible turbulent boundary layer.” J. Aerosp. Sci., 27共9兲, 675–678. Watters, G. Z., and Rao, M.共1971兲. “Hydrodynamic effect of seepage on

bed particles.” J. Hydr. Div., 97共3兲, 421–439.

Willetts, B. B., and Drossos, M. E.共1975兲. J. Hydr. Div., 101共12兲, 1477– 1488.

Closure to “Effect of Seepage-Induced

Nonhydrostatic Pressure Distribution

on Bed-Load Transport and Bed

Morphodynamics” by Simona Francalanci,

Gary Parker, and Luca Solari

April 2009, Vol. 134, No. 4, pp. 378–389.

DOI: 10.1061/共ASCE兲0733-9429共2008兲134:4共378兲

Simona Francalanci1; Gary Parker2; and Luca Solari3 1_{Postdoctoral Researcher, CERAFRI, Center of Research and Advanced}

Education for Hydrogeological Risk Prevention, Via XI Febbraio 2, 55040 Retignano di Stazzema 共LU兲, Italy. E-mail: simona. francalanci@dicea.unifi.it

2_{Prof., Ven Te Chow Hydrosystems Laboratory, Univ. of Illinois, 205 N.}

Mathews Ave., Urbana, IL 61801. E-mail: parkerg@uiuc.edu

3_{Asst. Prof., Dept. of Civil and Environmental Engineering, Univ. di}

Firenze, Via S. Marta 3, 50139 Firenze, Italy. E-mail: luca.solari@ dicea.unifi.it

The writers deeply appreciate this thoughtful discussion, which was also quite thought-provoking 共see below兲. The discussers

offer an encyclopedic overview of research on the effect of seep-age forces on sediment motion. The many papers they cite high-light the value of research in this area.

We begin by enumerating the ways in which our experiments differ from previous experimental work cited by the discussers. Martin共1970兲 considered incipient motion for the case of unidi-rectional pipe flow. The experiments of Baldock and Holmes 共1999兲 include both the cases of unidirectional flow and wave motion with a free surface. In the case of unidirectional flow, however, only the condition of incipient motion was considered. The experiments of Cheng and Chiew 共1999兲 likewise pertain only to incipient motion. Our experiments extend this work to the case of a fully mobile bed.

As far as we know, our experiments are the first to treat the effect of seepage on active sediment transport under sustained, unidirectional flow. The experiments were run for 12– 16 h to establish mobile-bed equilibrium in the absence of a zone of seep-age. After taking measurements, they were then continued for 2 – 4 h more to establish mobile-bed equilibrium in the presence of a zone of seepage. Measurements were taken both 10 min after the commencement of seepage and at mobile-bed equilibrium.

The results are unambiguous. In our experiments, upward seepage caused erosion in the seepage zone, ultimately resulting in a depression at mobile-bed equilibrium. Downward seepage caused deposition, ultimately resulting in an elevated zone at mobile-bed equilibrium. The scour can only have resulted from a tendency for enhanced sediment transport in the presence of upward seepage, and the fill can only have resulted from the opposite tendency in the presence of downward seepage. We be-lieve that our data constitute a benchmark for testing theoretical frameworks of the type outlined in our paper and the two papers cited by the discussers, Nielsen共1998兲 and Baldock and Holmes 共1999兲.

Having said this, we recognize that the discussers have de-tected an inconsistency in our own theoretical treatment. Specifi-cally, the formulation leading to Eq. 共16兲 of our paper is inconsistent with the formulation leading to Eq.共32兲 of the same paper. The writers accept the discussers’ contention that the physi-cal basis for the latter formulation, which is due to Cheng and Chiew共1999兲, is the correct one.

In the context of the analysis presented in our paper, the cor-rected form is ␶_*= ␶b

再

␳s−␳

冋

1 + vs K共1 − ␭p兲

册

冎

gD 共1兲

The discussers point out that they and their coauthors have pre-viously proposed forms similar to Eq. 共1兲. Specifically, Nielsen 共1998兲 proposed a form 关Eq. 共9兲 therein兴 which in the present notation takes the form

␶_*= ␶b

冋

␳s−␳

冉

1 +␤

vs K

冊

册

gD

共2兲

The parameter ␤, which Nielsen 共1998兲 approximated as 0.5, is intended to account for a possible suppression in the seepage force on particles exposed at the bed. Baldock and Holmes共1999兲 have proposed a form 关Eq. 共5兲 therein兴 which is identical to Eq.共1兲 above. These papers should have been cited in our paper. The writers acknowledge a lack of diligence in failing to track them down in the relevant conference proceedings.

The form of the modified Shields number proposed by Nielsen 共1998兲 evidently contains the same error as that of Eq. 共16兲 of our paper. The discussers, however, usefully incorporate the suppres-sion coefficient␤ into a modified form of Eq. 共1兲

␶_*= ␶b

再

␳s−␳

冋

1 + ␤vs K共1 − ␭p兲

册

冎

gD 共3兲

The above equation corresponds to Eq. 共4兲 of the discussers’. Using the porosity␭p= 0.30 pertaining to the experiments of our paper and a value of␤ of 1 共Baldock and Holmes 1999兲, Eq. 共3兲 indicates that the term vs/K of our Eq. 共16兲 should be corrected to 1.43 vs/K. Using the same value of ␭pand a value of␤ of 0.5 关as suggested by Nielsen共1998兲 based on Martin 共1970兲兴, the term vs/K should be corrected to 0.71 vs/K, while a value of ␤ of 0.7 would correspond to no correction. Thus while the derivation of Eq.共16兲 of our paper does indeed contain a nontrivial error which we hereby correct, the numerical predictions using such equation appear to be within the range of uncertainty suggested by the discussers.

We have used the example of Runs 1–3 共upward seepage兲 of our paper to test the effect of amending Eq.共16兲 of our paper to Eq.共3兲 in the numerical model. We know that the value ␤=0.7 recovers our original analysis. With this in mind, we have per-formed numerical runs with␤=0.5 and 1.0 as well. Fig. 1 shows

Fig. 1. Comparison of experimental and numerical results, Runs 1–3:

short-term experiment

Fig. 2. Comparison of experimental and numerical results, Runs 1–3:

the results of these numerical runs, along with the data for the short-term experiment. Fig. 2 shows the corresponding plot for the long-term experiment. The numerical results for the case ␤ = 1.0 somewhat overpredict the tendency for scour in the presence of upward seepage, suggesting the adoption of a value higher than 0.5 but lower than 1.0. The numerical runs with all three values of ␤, however, agree with the sense of the data; upward seepage produced enhanced sediment transport, resulting in scour.

The delineation of Eq.共3兲 should properly be considered the achievement of Nielsen共1998兲 and Baldock and Holmes 共1999兲. Here we attempt to counter their pessimism concerning the struc-ture of their own共and our兲 formulations, as expressed in the fol-lowing quote. “We therefore propose that the present treatment of this problem by关such equation兴 is neither particularly convincing with regards the extensive experimental data, nor easily justified on the basis of the fluid physics.”

The pessimism of the discussers seems to derive from two considerations:共1兲 the observation that the available data display disparate tendencies which may not be easily captured with frameworks that use relations of the type of Eq. 共3兲; and 共2兲 doubts concerning whether or not seepage forces affect the dy-namics of particles participating in bed-load transport.

We begin by considering the second consideration mentioned. The discussers state, with regard to the threshold of motion, the following: “Baldock and Holmes共1999兲 performed experiments with . . . both strong suction and injection, and observed no sig-nificant difference in the steady flow discharge required for in-cipient motion, consistent with much of previous experimental data cited above.” We believe that Cheng and Chiew共1999兲 have provided a physically justifiable theory of incipient motion in the presence of seepage effects, and have justified it extensively via experiments. It is their formulation that we have used in our framework. We believe that we are on solid ground here.

Also in regard to consideration two, the discussers state as follows: “once a particle lifts out of its bed recess and starts moving, it no longer experiences the seepage force that only acts within the bed matrix.” The implication is that seepage forces should not significantly affect bed-load transport itself. We offer the following counterargument. The volume bed-load transport rate per unit width qbcan be expressed as

qb= EbLstep 共4兲

where Ebdenotes the volume rate of entrainment of bed particles into bed-load per unit area and Lstep denotes the average step length of a particle. Extensive research has shown that while Eb

varies strongly with Shields number, Lstepshows a much weaker dependence 共Fernández Luque and van Beek 1976; Wong et al. 2007兲. Entrainment of a bed particle is a function of the forces at the bed, not after it has begun motion. With this in mind, we expect that seepage should enter into the physics of bed-load transport via the parameter Eb.

It may be that the effect of seepage on the parameter␶_* char-acterizing the Shields number of particles participating in motion is somewhat reduced compared to its effect on␶_*cr. Such a ten-dency may indeed be reflected in Figs. 1 and 2. That is, in the simulation using the suppression factor ␤=0.7 共which recovers our original calculation and fits the data well兲 in Eq. 共3兲 for ␶_*, we have not used a similar suppression factor␶_*crin the relation for the threshold of motion of Cheng and Chiew共1999兲, instead using their original relation.

Before proceeding to consideration one, we elaborate on the following statement of the discussers: “The authors ignore the possible influence of the seepage in directly modifying the bed shear stress in their model. However, very importantly, they in-clude the continuity effect.” We believe the two formulations amount to共more or less兲 the same thing, so that the incorporation of both would be double bookkeeping. Cheng and Chiew共1998a兲 offer a formulation of the type suggested by the discussers, Cheng and Chiew共1998b兲 offer the same formulation used in our paper, and Cheng and Chiew共1999, p. 674兲 indicate that the two meth-ods show good agreement.

We now proceed to consideration one, the fact that disparate results obtained to date cast doubts on formulations of the type in our paper. The constructive comments of the discussers motivated us to address the issue of disparate results. In doing this we use precisely the framework presented in our paper, with the excep-tions that共1兲 Eq. 共3兲 is used instead of the corresponding form in that paper, and共2兲 form drag is neglected in the bed-load trans-port equation. We return to the issue of form drag and bed forms below.

The bed-load transport is evaluated with Eq.共24兲 of our paper, neglecting form drag. This relation indicates that sediment mobil-ity in the presence of seepage is enhanced if seepage increases the excess Shields number above the critical value, and vice versa. Linearized perturbation analysis of our framework in the limit of small vs, applied to a bed that has not yet had time to respond morphodynamically to seepage, allows us to estimate conditions for enhanced bed mobility共mediated by dominant buoyancy ef-fects兲 versus suppressed mobility 共mediated by dominant suppres-sion of bed shear stress兲.

Our results are encapsulated in terms of a dimensionless pa-rameter, which we term the seepage number, defined as follows

Se = KL

UaHa

共5兲 In the above relation and subsequent relations, the subscript “a” refers to antecedent conditions in the absence of seepage.

For the sake of clarity we focus on upward seepage. The main result can be summarized as follows. When Se⬍Semin, where

Se_min is an order one quantity, buoyancy effects dominate, and mobility is enhanced everywhere in the seepage zone. As Se ⬎Semin, a subzone of suppressed mobility due to bed shear stress reduction appears from the upstream end of the seepage zone. This subzone extends downstream as Se increases, eventually reaching a value of at least共7/10兲L. The relation for Seminis as follows:

Fig. 3. Se_minas a function of␶

Semin= KLmin UaHa = 共M1␤ + M2R兲 RM1共1 − ␭p兲

冉

7 3X − 14 15冊 共6兲 where X =1 + Fra 2 1 − Fra2 共7a兲 M1= ␶_*a ␶_*a−␶_*cr,a , 共7b兲 M2= ␶_*cr,a ␶_*a−␶_*cr,a 共7c兲 R =␳s ␳ − 1 共7d兲

and Fradenotes the Froude number of the ambient flow. Compu-tations indicate that in general Seminis an order-one parameter. Fig. 3 shows a plot of Semin as a function of ␶_*a/␶_*cr,a for the values Fra= 0.2, 0.4, 0.6 and 0.8. In the plots, it has been assumed that R = 1.65共quartz兲, ␤=0.7 and ␭p= 0.3; that is, values applying to our experiments.

This analysis can explain conditions under which upward seepage results in either enhanced or suppressed mobility. We checked the above analysis against all 29 of our own experiments pertaining to ambient conditions 共no seepage兲. In these experi-ments, Fra ranged from 0.61 to 0.74. Values of Se computed for our experiments ranged from 0.08 to 0.171; values of Semin computed with Eq. 共6兲 ranged from 0.14 to 0.27. In all 29 cases

Se/Se_minwas less than unity. That is, our theory applied at the linear level indicates that upward seepage would cause enhanced mobility, as we observed.

Our computed values of Se/Seminranged from 0.52 to 0.79. If

Se/Se_min⬎1, Eq. 共6兲 indicates that a subzone would appear

within which mobility would be suppressed rather than enhanced by upward seepage. For sufficiently large values of Se/Semin, mo-bility suppression would dominate over the greater portion of the zone of seepage. According to Eq. 共6兲, this could have been achieved by e.g., increasing the length of the zone of seepage L or the hydraulic conductivity K.

To summarize, the same theoretical framework can explain disparate tendencies. It may turn out that this framework remains incomplete, but we are more optimistic than the discussers in thinking that it offers a roadmap for future research.

A final point concerns bedforms. As can be seen from Figs. 3 and 4 of our paper, the bedforms were of low amplitude, and their form drag never contributed more that 20% of the bed resis-tance. Detailed bed profiles were taken during the experiments 共Francalanci 2006兲; these did not indicate that seepage had a major effect on them. We feel it unlikely that the presence of bedforms is the cause of the enhanced/suppressed mobility in-duced by upward/downward seepage. This point, however, merits verification by means of experiments under conditions of lower-regime plane-bed.

We thank the discussers for their highly stimulating observa-tions. We also thank Yee-Meng Chiew, who greatly helped us in our thinking about the reasons for disparate results.

References

Baldock, T. E., and Holmes, P. 共1999兲. “Seepage effects on sediment transport by waves and currents.” Proc., 26th Int. Conf. on Coastal

Engineering, ASCE, Reston, Va., 3601–3614.

Cheng, N. S., and Chiew, Y. M.共1998a兲. “Modified logarithmic law for velocity distribution subjected to upward seepage.” J. Hydraul. Eng.,

124共12兲, 1235–1241.

Cheng, N. S., and Chiew, Y. M.共1998b兲. “Turbulent open-channel flow with upward seepage.” J. Hydraul. Res., 36共3兲, 415–431.

Cheng, N. S., and Chiew, Y. M.共1999兲. “Incipient sediment motion with seepage.” J. Hydraul. Res., 37共5兲, 665–681.

Fernández Luque, R., and van Beek, R.共1976兲. “Erosion and transport of bed-load sediment.” J. Hydraul. Res., 14共2兲, 127–144.

Francalanci, S. 共2006兲. “Sediment transport processes and local scale effects on river morphodynamics.” Ph.D. thesis, Univ. of Padova, Padova, Italy.

Martin, C. S.共1970兲. “Effect of a porous sand bed on incipient sediment motion.” Water Resour. Res., 6共4兲, 1162–1174.

Nielsen, P.共1998兲. “Coastal groundwater dynamics.” Coastal dynamics

’97, ASCE, Reston, Va., 546–555.

Wong, M., Parker, G., De Vries, P., Brown, T., and Burges, S. J.共2007兲. “Experiments on dispersion of tracer stones under lower-regime plane-bed equilibrium bed load transport.” Water Resources Research, 43共3兲, W03440.

Discussion of “Coherent Structures in

the Flow Field around a Circular Cylinder

with Scour Hole” by G. Kirkil, S. G.

Constaninescu, and R. Ettema

May 2008, Vol. 134, No. 5, pp. 572–587.

DOI: 10.1061/共ASCE兲0733–9429共2008兲134:5共572兲

Christian Gobert1; Oscar Link2; Michael Manhart3; and Ulrich Zanke4

1

Ph.D. Candidate, Fachgebiet Hydromechanik, Technische Universität München, Arcisstr. 21, 80333 München, Germany. E-mail: ch.gobert@bv.tum.de

2_{Asst. Prof., Dept. of Civil Engineering, Univ. of Concepción, Edmundo}

Larenas 270, Concepción, Chile. E-mail: olink@udec.cl

3

Prof., Fachgebiet Hydromechanik, Technische Universität München, Ar-cisstr. 21, 80333 München, Germany. E-mail: m.manhart@bv.tum.de

4

Prof., Institute for Hydraulic and Water Resources Engineering, Darmstadt Univ. of Technology, Rundeturmstraße 1, 64283. Germany. E-mail: wabau@wb.tu-darmstadt.de

Kirkil et al. 共2008兲 are complimented on their study involving both laboratory experiment and large eddy simulation 共LES兲, which contribute significantly to the understanding of scour mechanisms around a bridge pier. Recently, the discussers studied the interaction between flow field and sediment bed at a circular cylinder in sand by similar experiments and obtained comparable results that serve to extend the authors’ conclusions.

The authors presented laboratory-flume visualizations and re-sults of LES to investigate coherent structures present in the flow field around a circular cylinder located in an equilibrium scour hole in sand. By a remarkable analysis of the flow in the scour hole, they found a system of vortices共horseshoe vortex system, HV兲 more complex than hitherto indicated in literature.

In the discussers’ work bathymetry and flow at slightly differ-ent conditions than those of the authors were investigated. The Reynolds number based on bulk velocity and flow depth was

Re_b= 90,000, five times larger than in the authors’ work, based on pier diameter it was Rep= 60,000. The critical friction velocity ratio was 0.95. The sand bed grain sizes ranged between 0.6 and 2.0 mm, with a mean diameter of 0.97 mm. For the simulations, the discussers discretized the flow with up to 213 Mio. control volumes. For further details on the discussers’ laboratory and nu-merical experiments please refer to Link 共2006兲 and Link et al. 共2008a, b兲.

In the present discussion, the discussers highlight three points on the interaction between the HV and the scour-hole geometry that were not treated by the authors. These concern 共1兲 bathym-etry; 共2兲 layout and dynamics of the HV; and 共3兲 effects of time averaging.

Bathymetry

The discussers observed a similar final bathymetry as depicted in the authors’ Figs. 3, 4, and 6. In Fig. 1 are plotted profiles of the

sediment bed after 1, 6, 10, and 21 h on azimuthal half-planes with␪=0 and 90°. Three regions can be distinguished: a ring-shaped portion close to the cylinder base, a steep slope in the deeper part of the scour hole, and a mild slope in the upper part of it. Both slopes are separated by a clearly distinguishable kink in the bed 共Fig. 1兲. The shape of the scour hole remained nearly constant during scour development. Below the kink, the slope of the bed was measured to be 42°, while above the kink, the slope was 29°. The overall average slope was 35.5°, which is about 15% higher than the natural repose angle of the sand共30°兲. The observed scour-slopes suggest that the HV consist of at least three vortices of different intensities.

Layout and Dynamics of HV

Fig. 2 shows instantaneous fields from the discussers’ numerical experiment, similar to the authors’ Fig. 4. Three to four vortices can be distinguished. As indicated in the figure, two vortices V1 and V2 are found close to the pier. From animations the discuss-ers found that the most stable vortex is V2. This vortex is trapped

Fig. 1. Left and center: developing scour holes on the azimuthal half-planes with␪=0° and ␪=90°. Right: photo of the scour hole

in the scour hole and is observed at almost every instant whereas V1 cannot always be observed. Upstream of V2, a complex pat-tern of strongly intermittent vortices can be found. In comparison, the authors found three stable vortices in the scour hole. Taking into consideration that in the authors’ work the Reynolds number is smaller than in the discussers’ work, it can be concluded that frequency of vortex shedding and intermittency of vortices in-creases with Reynolds number.

Effects of Time Averaging

The discussers have also investigated differences between instan-taneous and time-averaged flow fields. This is of practical rel-evance since for cases in which time-averaged fields provide meaningful results, Reynolds averaged simulations共RANS兲 with appropriate closure models are a computationally less expensive alternative to LES for flow analysis. The authors found that the time-averaged flow field shows a similar vortex system as instan-taneous flow. In contrast, the discussers observed in the averaged and instantaneous fields two and up to four vortices respectively 共Fig. 2兲. This provides evidence that the leftmost vortex is not stable. One might conclude that this vortex is only an artifact of averaging because the instantaneous fields showed that in this region frequent vortex shedding and detachment takes place, a process that is hidden by time averaging. Thus, depending on the actual configuration, results from RANS might be misleading.

The discussers conclude that shedding, dissipation, and inter-action of vortices in the scour hole depends on the Reynolds number. Averaged flow fields and the corresponding bed shear stresses alone are not able to completely explain the scour mecha-nism at a cylinder. Based on the discussers’ observations and those of the authors, the discussers schematically depicted the scour mechanism in Fig. 2共bottom right兲. Close to the cylinder, scouring is controlled by the vortex V1, which appears only in-termittently. Farther upstream, a very stable counterrotating vor-tex V2 is observed. V2 rotates such that the flow near the bed points upstream whereas the flow near the bed below V1 points downstream. This explains the shape of the bed in these sections: at V1, the bed is flat because V1 transports sediment downstream, whereas at V2 the inclination of the bed is higher than the repose angle because V2 transports upstream and stabilizes the slope. At instances when V2 weakens, sediment slides down and is fed into V1.

Acknowledgments

Financial support of the Chilean Research Council, CONICYT, through grant 11080126 is acknowledged.

References

Kirkil, G., Constantinescu, S., and Ettema, R.共2008兲. “Coherent struc-tures in the flow field around a circular cylinder with scour hole.” J.

Hydraul. Eng., 134共5兲, 572–587.

Link, O.共2006兲. “An investigation on scouring around a single cylindri-cal pier in sand.” Mitteilungen des Institutes für Wasserbau und Wasserwirtschaft der Technischen Universität Darmstadt, Nr. 136共in German兲.

Link, O., Gobert, Ch., Manhart, M., and Zanke, U.共2008a兲. “Effect of the horseshoe vortex system on the geometry of a developing scour hole

at a cylinder.” Proc. 4th Int. Conf. on Scour and Erosion.

Link, O., Pfleger, F., and Zanke, U.共2008b兲. “Characteristics of develop-ing scour-holes at a sand-embedded cylinder.” Int. J. Sediment Res.,

23, 268–276.

Closure to “Coherent Structures in the

Flow Field around a Circular Cylinder with

Scour Hole” by G. Kirkil, S. G.

Constantinescu, and R. Ettema

May 2008, Vol. 134, No. 5, pp.572–587.

DOI: 10.1061/共ASCE兲0733-9429共2009兲134:5共572兲

G. Kirkil, M.ASCE1; S. G. Constantinescu, M.ASCE2; and R. Ettema, M.ASCE3

1

Graduate Research Asst., Civil & Environmental Engineering Dept., IIHR-Hydroscience and Engineering, The Univ. of Iowa, Stanley Hy-draulics Laboratory, Iowa City, IA 52242; currently, Postdoctoral Re-search Staff Member, Atmospheric, Earth and Energy Division, Lawrence Livermore National Lab, P.O. Box 808, L-103, Livermore, CA 94551. E-mail: gokhan-kirkil@uiowa.edu; kirkil1@llnl.gov

2

Assoc. Prof., Civil & Environmental Engineering, IIHR-Hydroscience and Engineering, The Univ. of Iowa, Stanley Hydraulics Laboratory, Iowa City, IA 52242. E-mail: sconstan@engineering.uiowa.edu

3_{Prof., College of Engineering and Applied Science, The Univ. of}

Wyo-ming, Laramie, WY 82071. E-mail: rettema@uwyo.edu

The writers wish to thank the discussers for their kind words and comments. The discussers are commended for bringing out inter-esting issues in relation to large eddy simulation共LES兲 of flow and turbulence structure at circular cylinders in loose bed chan-nels, in particular related to the effects of Reynolds number and bathymetry shape. We think the fact that the main features of the flow fields are to a large degree similar in two LES investigations conducted with different numerical codes gives additional confi-dence in the approach chosen by the writers and the discussers. Several specific comments of the discussers are addressed below. Our comments are also based on additional results 共see Kirkil, Constantinescu, and Ettema 2009b; Kirkil and Constantinescu 2009a; Koken and Constantinescu 2009兲 obtained by our group using detached eddy simulation 共DES兲. In particular, the DES simulation of Kirkil, Constantinescu, and Ettema 共2009b兲 was performed at a cylinder Reynolds number of 2.06ⴱ105_{. The DES} simulation further extends the range of Reynolds numbers needed to understand and quantify scale effects on the flow and turbu-lence structure. The Reynolds number of the numerical investiga-tion conducted by the discussers falls in between共ReD= 60,000, D is the cylinder diameter兲.

Fig. 1. Mean flow streamlines in the␾=0° plane a兲 Re_D= 16,000;共b兲 ReD= 2.06ⴱ105

The change in the bed slope within the scour hole observed by the discussers was documented by several experiments 共Unger and Hager 2007兲. The difference between the two slopes tends to decrease as the bed evolves toward equilibrium. In our experi-ments, including in the ones performed at ReD= 2.06ⴱ105, the change in the bed slope within the scour hole was found to be relatively small. Despite this, the mean flow topology at ReD = 16,000 and ReD= 2.06ⴱ105 as visualized by 2D streamlines 共Fig. 1兲 was found to be similar within the scour hole, especially near the symmetry plane共polar angle ␾=0°兲, and to contain two main necklace vortices that can be associated with V2 and V3 in the mean flow results presented by the discussers 共Fig. 2 of the discussion兲.

We fully agree with the conclusion by the discussers that the vortical system inside the scour hole consists of at least three vortices of different intensities共two main necklace vortices, one junction vortex and, depending on the bathymetry and other flow conditions, a number of bottom wall-attached vortices induced by the presence of the main necklace vortices兲. A difference observed in the mean flow patterns in the symmetry plane between the simulations performed by the authors and the discussers is the presence in our simulation at ReD= 16,000 of a bottom-attached vortex between the main two necklace vortices and especially of a small, but strongly coherent, junction vortex at the base of the cylinder at both Reynolds numbers. Though the differences in the flow conditions and bathymetry explain the absence of the bottom-attached vortex in the mean flow patterns predicted in the simulation conducted by the discussers and in our simulation at ReD= 2.06ⴱ105关Fig. 1共b兲兴, the absence of a well-defined junction vortex is more surprising, given the fact that experiments have always shown the formation of that vortex, which is in fact re-sponsible for the slight increase of the bathymetry levels very close to the cylinder. Though a small vortex denoted V1 is ob-served in the instantaneous flow fields predicted by the discussers 共Fig. 2 in the discussion兲, that vortex is in fact a bottom-attached vortex induced by the main necklace vortex V2 rather than a true junction vortex as shown in the sketch in the same figure. In the simulation performed by the discussers a junction vortex is not present in the mean flow field in the ␾=0° plane despite the presence of a well-defined downflow with streamlines deflected toward the cylinder as the bed is approached. The fact that in the instantaneous flow the 2D streamlines close to the cylinder are not always tangent to the cylinder surface suggests a problem related to the resolution of the flow in that region. Comparison of our simulations at ReD= 16,000 and ReD= 2.06ⴱ105共for more de-tails, see Kirkil, Constantinescu, and Ettema 2009b兲 shows that a main scale effect is the reduction of the lateral extent of the up-stream necklace vortex V3 with the increase in the Reynolds number. For example, that vortex is still present in the ␾=45° section at ReD= 16,000 but absent for 兩␾兩⬎30° in the ReD = 2.06ⴱ105_simulation.

Related to the unsteady dynamics of the necklace vortices, our results at both Reynolds numbers are fully consistent with those by the discussers in the sense that the vortex denoted V2 in their discussion共main necklace vortex in our paper兲 is the most coher-ent and stable vortex while, at times, the coherence of the main secondary vortex V3 is lost and the region is populated by a large number of smaller eddies. In fact, the instantaneous flow structure visualized in Fig. 4 of our paper is fully consistent with this scenario. One important point, already mentioned in the original article is that the presence of a certain number of vortices in the mean flow does not necessarily mean that they have a direct cor-respondent in the instantaneous flow fields or that they are stable, in the sense that they can be recognized in each of the

instanta-neous flow fields. As the discussers correctly point out, that is the case for V2 and, based on our results, for the junction vortex but not for the primary secondary vortex V3 or for the small vortices shed from the separating region of the incoming boundary layer. We do not think V3 is an artifact of the averaging. This is because such a vortex is present at small polar angles over more than 50% of the time in our simulations. However, depending on the shape of the scour hole, this is a distinct possibility especially during the initial stages of the development of the scour hole. In this regard, the comment by the discussers that results from RANS might be misleading to understand the structure of the HV system in the instantaneous flow fields is a very important one. For example, a similar problem arises when the structure of the detached shear layers on the sides of the cylinder is analyzed based on RANS or even unsteady RANS results. Unsteady RANS simulations do not capture the vortex tubes shed in the separated shear layers, which play a determinant role in determining the turbulence structure of the near wake region despite capturing the large-scale shedding of the rollers.

We do not think comparison of Figs. 4 and 6 in our paper suggests the time-averaged flow fields show a similar vortex sys-tem as the instantaneous flow at ReD= 16,000. In fact, we fully agree with the discussers that is not the case in the region between the separation line of the incoming boundary layer and the main necklace vortex V2. Animations of the instantaneous flow pat-terns from the solutions obtained by the authors support that ob-servation at both Reynolds numbers. We also agree that the Reynolds number has an influence on the dynamics of the horse-shoe vortex system inside the scour hole. However, comparison of the LES and DES simulations 共Kirkil, Constantinescu, and Ettema 2009b兲 performed by the writers clearly shows that the main necklace vortex is subject to bimodal oscillations at both Reynolds numbers. The large-scale bimodal oscillations of this necklace vortex are the main reason for the large amplification of the turbulence within the scour hole in front of the cylinder. An-other very relevant result related to scale effects is that the in-crease in the Reynolds number dein-creases the amplitude of the bimodal oscillation of the main necklace vortex V2. The discuss-ers’ comment that the intermittency of the vortices increases with the Reynolds number is consistent with our DES results.

We fundamentally agree with the discussion of the scour mechanism by the discussers. We have two comments based on our LES and DES results. The first is that in our simulations the junction vortex is quite stable and strong, consistent with most experimental investigations. Still, large-scale temporal variations in its strength are observed, especially during the times the main necklace vortex approaches the face of the cylinder or a large eddy is convected with the downflow parallel to the face of the cylinder. We would also like to point out that the formation of a horseshoe vortex system containing two main necklace vortices 共V2 and V3 in the notation used by the discussers兲 does not nec-essarily imply the bathymetry inside the upstream part of the scour hole should display a large change in the bed slope in be-tween the two vortices.

References

Kirkil, G., and Constantinescu, G.共2009a兲. “Nature of flow and turbu-lence structure around an in-stream vertical plate in a shallow channel and the implications for sediment erosion.” Water Resour. Res., 44, W06412.

Kirkil, G., Constantinescu, G., and Ettema, R.共2009b兲. “DES investiga-tion of turbulence and sediment transport at a circular pier with scour

hole.” J. Hydraul. Eng., 135共11兲, 888–901.

Koken, M., and Constantinescu, G.共2009兲. “An investigation of the dy-namics of coherent structures in a turbulent channel flow with a ver-tical sidewall obstruction.” Phys. Fluids, 21, 085104.

Unger, J., and Hager, W. H. 共2007兲. “Downflow and horseshoe vortex characteristics of sediment embedded bridge piers.” Exp. Fluids,

42共1兲, 1–19.

Discussion of “Inception Point and Air

Concentration in Flows on Stepped Chutes

Lined with Wedge-Shaped Concrete

Blocks” by Antònio T. Relvas and Antònio

N. Pinheiro

August 2008, Vol. 134, No. 8, pp. 1042–1051.

DOI: 10.1061/共ASCE兲0733-9429共2008兲134:8共1042兲

Sanjay W. Thakare1and Sandip P. Tatewar2 1

Lecturer, Dept. of Civil Engineering, Government College of Engineer-ing, Amravati. 444 604, India. E-mail: thakare.sanjay@gcoea.ac.in

2_{Asst. Prof., Dept. of Civil Engineering, Government College of}

Engi-neering, Amravati, 444 604, India. E-mail: sandiptatewar@ yahoo.co.in

The authors are to be congratulated for carrying out a systematic investigation of flow over a stepped chute lined with wedge-shaped concrete blocks 共hereafter called a stepped block spill-way兲. As mentioned by the authors, Pinheiro and Revels 共2000兲 showed that the channel excavated in the embankment and pro-tected with wedge-shaped precast concrete blocks can offer an

inexpensive alternative to conventional concrete-cast in-situ solu-tions. The discussers carried out a case study of Chandrabhaga River Project in the Maharashtra State of India, for which a con-ventional ogee spillway is provided. Thakare et al. 共2008兲 de-signed a stepped block spillway following CIRIA design guidelines共Hewlett et al. 1997兲. In this project, it was found that the cost required for a stepped block spillway was considerably less than that of the existing ogee spillway. The stepped block spillway has, therefore, considerable potential as a low cost method for providing a spillway and the protection of embank-ment from erosion by overtopping flow. Hence, the discussers support the authors’ statement that stepped-block spillways are a promising solution to provide overtopping protection for embank-ment dam if the discharge capacity of the existing spillway is not adequate.

Stepped-block spillways are generally associated with em-bankment dams for overtopping protection and hence designed for flatter slopes共hs/ls= 0.25 to 0.50兲, whereas stepped spillways are generally associated with concrete dams and designed for steeper slopes共hs/ls= 1.25 to 1.66兲. Fig. 1 shows that the differ-ence between the values of hc/hspredicted by equations关Chanson 共1996兲 and Andre 共2004兲兴 and experimental values for gabion-stepped spillway is large for flatter slopes as compared to that for steeper slopes. The authors compared the characteristics of flow 共like onset of skimming flow, location of inception point, and air concentration兲 over a stepped-block spillway with those of a stepped spillway for roller compacted concrete dams. In the latter case, the steps are impervious, whereas a stepped-block spillway allows the percolation of flow through the joints between the blocks. A gabion-stepped spillway also permits infiltration of flow inside the rocks, which is similar in nature to the flow over stepped block spillway. This infiltration modifies the